In the field of digital communications researchers are constantly striving for methods to increase data transmission rate and reduce bit error probabilities (BEP). Traditionally, digital inputs are carried on a high frequency sinusoidal analog signal (called the carrier signal) via modulation by using a suitable mixing circuit. The resulting modulated signal is then transmitted through known means to a receiver. At the receiver, a mixing circuit demodulates the received signal by using a high frequency sinusoidal wave and a suitable filter to retrieve the originally transmitted digital input signals.
One problem encountered during the transmission process is that intersymbol interference (ISI) and Gaussian noise are added to the received signal which causes the received signal to differ from the transmitted signal, thereby causing transmission errors. Thus, the probabilities of error in the presence of ISI and Gaussian noise is dramatically increased.
Generally speaking, bit errors are calculated according to a following generalized example. Consider a word of M bits {b0, b1, . . . bM-1} to be mapped to a pulse series consisting of signals Skgk, where skε{−1, 0, +1} is the kth pulse coefficient and gk is the kth pulse. The result of mapping the M bits yields the pulse series {S0g(t), S1g(t−TB), . . . , SM-1g(t−(M−1)TB)}, where TB represents the bit time. If M=2 then the resulting probability of error is worse than the corresponding probability for a classical binary case (M=2) and each bit is mapped to either −g(t) or +g(t). The received signal at time t−iTB is expressed as:Y(t−iTB)=n+S1g(t−iTB)+ΣSkg(t−kTB) where (−∞<k<∞,k≠i)  (1)
where n represents the Gaussian additive noise and ΣSkg(t−kTB) from k=−∞ to ∞, with the exception of k=i represents the ISI term. Assume that Z represents the ISI, and X=n+Z, thenYi=Sigi+x  (2)
where t and TB are omitted for notational economy. For a received signal Yi, the probability of correct decision regarding pulse coefficient Si conditioned on Z can be expressed asPc|z=P(Si=−1)P(−gi+X<gi/2)+P(Si=0)P(−gi/2<X<+gi/2)+P(Si=+1)P(gi+X>gi/2)  (3)
Assume that X is symmetrical around zero [4], and using P(n+Z<+gi/2)=1−P(n+Z>+gi/2), and sinceP(−gi/2<n+Z<+gi/2)=1−P(n+Z<−gi/2)−P(n+Z>+gi/2)=1−2P(n+Z>+gi/2)  (4)andP(n+Z<+gi/2)=1−P(n+Z>+gi/2)  (5)
then Eq. (3) may be rewritten asPc|z=P(Si=−1)(1−p(N+z>gi/2))+P(si=0)(1−2p(n+Z>gi/2))+P(Si=+1)(1−P(n+Z>gi/2))  (6)
Equation (6) can be rewritten asPc|z=1−P(n+Z>gi/2)(P(Si=−1)+2P(Si=0)+P(Si=+1))=1=P(n>gi/2−Z)[P(Si=−1)+2P(Si=0)+P(Si=+1)]  (7)
Since the probability of error conditioned on Z is 1−Pc then,Pe(Si|z=G(gi/2−Z)[P(Si=−1)+2P(Si=0)+P(Si+1)]  (8)
where G(.) represents the cdf of the Gaussian noise. Assume that C1=[P(Si=−1)+2P(Si=0)+P(Si=+1)], and fz(z) represents the pdf of the random variable Z, then Pe(Si) can be expressed asPe(Si)=C1∫fz(Z)G(gi/2−Z)dZ  (9)
For M=2, any error in S0 or S1 will result in a symbol error. A symbol error may result either in two bit error or in one bit errors. Since the probability of having one error bit is twice the probability of having two error bits within a symbol once a symbol error occurs, then the average probability of bit error is ⅓Pe(Si).
One approach taken to minimize ISI in U.S. Pat. No. 7,257,163 to Hwang et al is to map a cluster digital words to a single mapped word. In Hwang et al, a digital input is mapped to one of a robust set of digital golden words. The golden words are much longer than the digital input. Therefore, even though the bit error rate is acceptable, the transmission rate of Hwang is rather slow.